
<A NAME=taylorcombine>

<TITLE>taylorcombine</TITLE></A>
<b><a href=r37_idx.html>INDEX</a></b><p><p>



<B>TAYLORCOMBINE</B> _ _ _  _ _ _  _ _ _  _ _ _ <B>operator</B><P>
<P>
 
 This operator tries to combine all Taylor kernels found in its 
 argument into one. Operations currently possible are: 
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<P>
 _ _ _ Addition, subtraction, multiplication, and division. 
 <P>
 _ _ _ Roots, exponentials, and logarithms. 
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 _ _ _ Trigonometric and hyperbolic functions and their inverses. 
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<P>
 <P> <H3> 
examples: </H3>
<P><PRE><TT>
    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    taylorcombine log hugo; 

         3
  X + O(X )


    taylorcombine(hugo + x); 

           1  2      3
  (1 + X + -*X  + O(X )) + X
           2


    on taylorautoexpand; 

    taylorcombine(hugo + x); 

            1  2      3
  1 + 2*X + -*X  + O(X )  
            2

</TT></PRE><P>Application of unary operators like <em>log</em> and <em>atan</em>
 
 will nearly always succeed. For binary operations their arguments 
 have to be Taylor kernels with the same template. This means that 
 the expansion variable and the expansion point must match. 
 Expansion order is not so important, different order usually means 
 that one of them is truncated before doing the operation. 
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<P>
If 
<A HREF=r37_0539.html>taylorkeeporiginal</A> is set to <em>on</em> and if all 
 Taylor kernels in its argument have their original expressions 
 kept <em>taylorcombine</em> will also combine these and store the 
 result as the original expression of the resulting Taylor kernel. 
 There is also the switch 
<A HREF=r37_0537.html>taylorautoexpand</A>. 
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<P>
There are a few restrictions to avoid mathematically undefined 
 expressions: it is not possible to take the logarithm of a Taylor 
 kernel which has no terms (i.e. is zero), or to divide by such a 
 beast. There are some provisions made to detect singularities 
 during expansion: poles that arise because the denominator has 
 zeros at the expansion point are detected and properly treated, 
 i.e. the Taylor kernel will start with a negative power. (This 
 is accomplished by expanding numerator and denominator separately 
 and combining the results.) Essential singularities of the known 
 functions (see above) are handled correctly. 
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<P>
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